(* Title : Fact.thy
Author : Jacques D. Fleuriot
Copyright : 1998 University of Cambridge
Conversion to Isar and new proofs by Lawrence C Paulson, 2004
The integer version of factorial and other additions by Jeremy Avigad.
*)
header{*Factorial Function*}
theory Fact
imports Main
begin
class fact =
fixes
fact :: "'a \<Rightarrow> 'a"
instantiation nat :: fact
begin
fun
fact_nat :: "nat \<Rightarrow> nat"
where
fact_0_nat: "fact_nat 0 = Suc 0"
| fact_Suc: "fact_nat (Suc x) = Suc x * fact x"
instance proof qed
end
(* definitions for the integers *)
instantiation int :: fact
begin
definition
fact_int :: "int \<Rightarrow> int"
where
"fact_int x = (if x >= 0 then int (fact (nat x)) else 0)"
instance proof qed
end
subsection {* Set up Transfer *}
lemma transfer_nat_int_factorial:
"(x::int) >= 0 \<Longrightarrow> fact (nat x) = nat (fact x)"
unfolding fact_int_def
by auto
lemma transfer_nat_int_factorial_closure:
"x >= (0::int) \<Longrightarrow> fact x >= 0"
by (auto simp add: fact_int_def)
declare transfer_morphism_nat_int[transfer add return:
transfer_nat_int_factorial transfer_nat_int_factorial_closure]
lemma transfer_int_nat_factorial:
"fact (int x) = int (fact x)"
unfolding fact_int_def by auto
lemma transfer_int_nat_factorial_closure:
"is_nat x \<Longrightarrow> fact x >= 0"
by (auto simp add: fact_int_def)
declare transfer_morphism_int_nat[transfer add return:
transfer_int_nat_factorial transfer_int_nat_factorial_closure]
subsection {* Factorial *}
lemma fact_0_int [simp]: "fact (0::int) = 1"
by (simp add: fact_int_def)
lemma fact_1_nat [simp]: "fact (1::nat) = 1"
by simp
lemma fact_Suc_0_nat [simp]: "fact (Suc 0) = Suc 0"
by simp
lemma fact_1_int [simp]: "fact (1::int) = 1"
by (simp add: fact_int_def)
lemma fact_plus_one_nat: "fact ((n::nat) + 1) = (n + 1) * fact n"
by simp
lemma fact_plus_one_int:
assumes "n >= 0"
shows "fact ((n::int) + 1) = (n + 1) * fact n"
using prems unfolding fact_int_def
by (simp add: nat_add_distrib algebra_simps int_mult)
lemma fact_reduce_nat: "(n::nat) > 0 \<Longrightarrow> fact n = n * fact (n - 1)"
apply (subgoal_tac "n = Suc (n - 1)")
apply (erule ssubst)
apply (subst fact_Suc)
apply simp_all
done
lemma fact_reduce_int: "(n::int) > 0 \<Longrightarrow> fact n = n * fact (n - 1)"
apply (subgoal_tac "n = (n - 1) + 1")
apply (erule ssubst)
apply (subst fact_plus_one_int)
apply simp_all
done
lemma fact_nonzero_nat [simp]: "fact (n::nat) \<noteq> 0"
apply (induct n)
apply (auto simp add: fact_plus_one_nat)
done
lemma fact_nonzero_int [simp]: "n >= 0 \<Longrightarrow> fact (n::int) ~= 0"
by (simp add: fact_int_def)
lemma fact_gt_zero_nat [simp]: "fact (n :: nat) > 0"
by (insert fact_nonzero_nat [of n], arith)
lemma fact_gt_zero_int [simp]: "n >= 0 \<Longrightarrow> fact (n :: int) > 0"
by (auto simp add: fact_int_def)
lemma fact_ge_one_nat [simp]: "fact (n :: nat) >= 1"
by (insert fact_nonzero_nat [of n], arith)
lemma fact_ge_Suc_0_nat [simp]: "fact (n :: nat) >= Suc 0"
by (insert fact_nonzero_nat [of n], arith)
lemma fact_ge_one_int [simp]: "n >= 0 \<Longrightarrow> fact (n :: int) >= 1"
apply (auto simp add: fact_int_def)
apply (subgoal_tac "1 = int 1")
apply (erule ssubst)
apply (subst zle_int)
apply auto
done
lemma dvd_fact_nat [rule_format]: "1 <= m \<longrightarrow> m <= n \<longrightarrow> m dvd fact (n::nat)"
apply (induct n)
apply force
apply (auto simp only: fact_Suc)
apply (subgoal_tac "m = Suc n")
apply (erule ssubst)
apply (rule dvd_triv_left)
apply auto
done
lemma dvd_fact_int [rule_format]: "1 <= m \<longrightarrow> m <= n \<longrightarrow> m dvd fact (n::int)"
apply (case_tac "1 <= n")
apply (induct n rule: int_ge_induct)
apply (auto simp add: fact_plus_one_int)
apply (subgoal_tac "m = i + 1")
apply auto
done
lemma interval_plus_one_nat: "(i::nat) <= j + 1 \<Longrightarrow>
{i..j+1} = {i..j} Un {j+1}"
by auto
lemma interval_Suc: "i <= Suc j \<Longrightarrow> {i..Suc j} = {i..j} Un {Suc j}"
by auto
lemma interval_plus_one_int: "(i::int) <= j + 1 \<Longrightarrow> {i..j+1} = {i..j} Un {j+1}"
by auto
lemma fact_altdef_nat: "fact (n::nat) = (PROD i:{1..n}. i)"
apply (induct n)
apply force
apply (subst fact_Suc)
apply (subst interval_Suc)
apply auto
done
lemma fact_altdef_int: "n >= 0 \<Longrightarrow> fact (n::int) = (PROD i:{1..n}. i)"
apply (induct n rule: int_ge_induct)
apply force
apply (subst fact_plus_one_int, assumption)
apply (subst interval_plus_one_int)
apply auto
done
lemma fact_mono_nat: "(m::nat) \<le> n \<Longrightarrow> fact m \<le> fact n"
apply (drule le_imp_less_or_eq)
apply (auto dest!: less_imp_Suc_add)
apply (induct_tac k, auto)
done
lemma fact_neg_int [simp]: "m < (0::int) \<Longrightarrow> fact m = 0"
unfolding fact_int_def by auto
lemma fact_ge_zero_int [simp]: "fact m >= (0::int)"
apply (case_tac "m >= 0")
apply auto
apply (frule fact_gt_zero_int)
apply arith
done
lemma fact_mono_int_aux [rule_format]: "k >= (0::int) \<Longrightarrow>
fact (m + k) >= fact m"
apply (case_tac "m < 0")
apply auto
apply (induct k rule: int_ge_induct)
apply auto
apply (subst add_assoc [symmetric])
apply (subst fact_plus_one_int)
apply auto
apply (erule order_trans)
apply (subst mult_le_cancel_right1)
apply (subgoal_tac "fact (m + i) >= 0")
apply arith
apply auto
done
lemma fact_mono_int: "(m::int) <= n \<Longrightarrow> fact m <= fact n"
apply (insert fact_mono_int_aux [of "n - m" "m"])
apply auto
done
text{*Note that @{term "fact 0 = fact 1"}*}
lemma fact_less_mono_nat: "[| (0::nat) < m; m < n |] ==> fact m < fact n"
apply (drule_tac m = m in less_imp_Suc_add, auto)
apply (induct_tac k, auto)
done
lemma fact_less_mono_int_aux: "k >= 0 \<Longrightarrow> (0::int) < m \<Longrightarrow>
fact m < fact ((m + 1) + k)"
apply (induct k rule: int_ge_induct)
apply (simp add: fact_plus_one_int)
apply (subst mult_less_cancel_right1)
apply (insert fact_gt_zero_int [of m], arith)
apply (subst (2) fact_reduce_int)
apply (auto simp add: add_ac)
apply (erule order_less_le_trans)
apply (subst mult_le_cancel_right1)
apply auto
apply (subgoal_tac "fact (i + (1 + m)) >= 0")
apply force
apply (rule fact_ge_zero_int)
done
lemma fact_less_mono_int: "(0::int) < m \<Longrightarrow> m < n \<Longrightarrow> fact m < fact n"
apply (insert fact_less_mono_int_aux [of "n - (m + 1)" "m"])
apply auto
done
lemma fact_num_eq_if_nat: "fact (m::nat) =
(if m=0 then 1 else m * fact (m - 1))"
by (cases m) auto
lemma fact_add_num_eq_if_nat:
"fact ((m::nat) + n) = (if m + n = 0 then 1 else (m + n) * fact (m + n - 1))"
by (cases "m + n") auto
lemma fact_add_num_eq_if2_nat:
"fact ((m::nat) + n) =
(if m = 0 then fact n else (m + n) * fact ((m - 1) + n))"
by (cases m) auto
subsection {* @{term fact} and @{term of_nat} *}
lemma of_nat_fact_not_zero [simp]: "of_nat (fact n) \<noteq> (0::'a::semiring_char_0)"
by auto
lemma of_nat_fact_gt_zero [simp]: "(0::'a::{linordered_semidom}) < of_nat(fact n)" by auto
lemma of_nat_fact_ge_zero [simp]: "(0::'a::linordered_semidom) \<le> of_nat(fact n)"
by simp
lemma inv_of_nat_fact_gt_zero [simp]: "(0::'a::linordered_field) < inverse (of_nat (fact n))"
by (auto simp add: positive_imp_inverse_positive)
lemma inv_of_nat_fact_ge_zero [simp]: "(0::'a::linordered_field) \<le> inverse (of_nat (fact n))"
by (auto intro: order_less_imp_le)
end